Selected topics from the original collection; this is not a complete chapter list.
CLASS 11 FORMULA VAULT
1. SETS
Basic Set Concepts
A ∈ B → A is an element of B
A ⊆ B → A is a subset of B
A ⊂ B → A is a proper subset of B
n(A) → Number of elements in set A
Types of Sets
Empty Set: ∅
Singleton Set: A set having exactly one element
Finite Set: n(A) is finite
Infinite Set: n(A) is infinite
Power Set
If n(A) = n, then
n(P(A)) = 2n
Number of Subsets
Number of subsets = 2n
Number of proper subsets = 2n − 1
Union and Intersection
A ∪ B → Elements belonging to A or B
A ∩ B → Elements common to A and B
A − B → Elements belonging to A but not B
Number of Elements
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Three Sets
n(A ∪ B ∪ C) = n(A) + n(B) + n(C)
− n(A ∩ B) − n(B ∩ C) − n(C ∩ A)
+ n(A ∩ B ∩ C)
De Morgan's Laws
(A ∪ B)' = A' ∩ B'
(A ∩ B)' = A' ∪ B'
2. RELATIONS AND FUNCTIONS
Cartesian Product
A × B = {(a,b) : a ∈ A, b ∈ B}
If n(A) = m and n(B) = n, then
n(A × B) = mn
Number of Relations
If n(A) = m and n(B) = n, then
Number of relations from A to B = 2mn
Function
f : A → B
Every element of A must have exactly one image in B.
Number of Functions
If n(A) = m and n(B) = n, then
Number of functions from A to B = nm
Composition of Functions
(f ∘ g)(x) = f(g(x))
(g ∘ f)(x) = g(f(x))
Identity Function
I(x) = x
f ∘ I = I ∘ f = f
4. PRINCIPLE OF MATHEMATICAL INDUCTION
Principle
To prove P(n) for all n ∈ N:
1. Prove P(1) is true.
2. Assume P(k) is true.
3. Prove P(k + 1) is true.
4. Therefore P(n) is true for all n ∈ N.
Induction Step
P(k) ⇒ P(k + 1)
5. COMPLEX NUMBERS AND QUADRATIC EQUATIONS
Complex Number
z = a + ib
where a,b ∈ R and i = √(−1)
Important Powers of i
i2 = −1
i3 = −i
i4 = 1
Conjugate
If z = a + ib, then
z̄ = a − ib
Modulus
|z| = √(a2 + b2)
Important Results
z + z̄ = 2a
z − z̄ = 2ib
z z̄ = |z|2
Quadratic Equation
ax2 + bx + c = 0, a ≠ 0
Quadratic Formula
x = (−b ± √(b2 − 4ac))/(2a)
Discriminant
D = b2 − 4ac
Nature of Roots
D > 0 → Two distinct real roots
D = 0 → Two equal real roots
D < 0 → No real roots
Sum and Product of Roots
α + β = −b/a
αβ = c/a
6. LINEAR INEQUALITIES
Basic Rules
If a < b, then
a + c < b + c
If a < b and c > 0, then
ac < bc
Sign Reversal
If a < b and c < 0, then
ac > bc
Interval Notation
a < x < b → (a,b)
a ≤ x ≤ b → [a,b]
a ≤ x < b → [a,b)
a < x ≤ b → (a,b]
7. PERMUTATIONS AND COMBINATIONS
Factorial
n! = n(n − 1)(n − 2) ... 3 × 2 × 1
0! = 1
Permutation
nPr = n!/(n − r)!
Combination
nCr = n!/[r!(n − r)!]
Important Results
nCr = nCn−r
nCr = n−1Cr + n−1Cr−1
Relation Between P and C
nPr = r! × nCr
Circular Permutations
Number of circular arrangements of n distinct objects
= (n − 1)!
8. BINOMIAL THEOREM
Binomial Expansion
(a + b)n = Σr=0n nCr an−rbr
General Term
Tr+1 = nCr an−rbr
Middle Term
If n is even → one middle term
Middle term = T(n/2)+1
If n is odd → two middle terms
T(n+1)/2 and T(n+3)/2
Important Results
nC0 + nC1 + ... + nCn = 2n
nC0 − nC1 + nC2 − ... = 0
9. SEQUENCES AND SERIES
Arithmetic Progression
a, a + d, a + 2d, ...
Common Difference
d = a2 − a1
nth Term of AP
an = a + (n − 1)d
Sum of n Terms of AP
Sn = n/2 [2a + (n − 1)d]
Sn = n/2(a + l)
Arithmetic Mean
AM = (a + b)/2
Geometric Progression
a, ar, ar2, ar3, ...
nth Term of GP
an = arn−1
Sum of n Terms of GP
Sn = a(rn − 1)/(r − 1), r ≠ 1
Sn = a(1 − rn)/(1 − r), r ≠ 1
Sum of Infinite GP
S∞ = a/(1 − r), |r| < 1
Geometric Mean
GM = √(ab)
Important Standard Sums
1 + 2 + 3 + ... + n = n(n + 1)/2
12 + 22 + 32 + ... + n2
= n(n + 1)(2n + 1)/6
13 + 23 + 33 + ... + n3
= [n(n + 1)/2]2
QUICK REVISION
Most Important Formulas
n(P(A)) = 2n
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Number of functions = nm
x = (−b ± √(b2 − 4ac))/(2a)
nPr = n!/(n − r)!
nCr = n!/[r!(n − r)!]
an = a + (n − 1)d
Sn = n/2[2a + (n − 1)d]
an = arn−1
S∞ = a/(1 − r), |r| < 1
Conditions: factorial and binomial formulas use non-negative integer n and 0 ≤ r ≤ n; the alternating binomial sum is zero for n ≥ 1. The real geometric mean √(ab) here is for non-negative a and b. Circular arrangements identify rotations, not reflections.